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Time Delayed Feedback Control Applied in an Atomic Force Microscopy (AFM) Model in Fractional-Order

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Abstract

Purpose

In this work, the nonlinear dynamics and control of an Atomic Force Microscopy (AFM) model in fractional-order were investigated.

Methods and Results

For analyse of the chaos behaviour, the 0–1 test was used, since this is a good tool to characterise fractional-order differential systems. To bring the system from a chaotic state to a periodic orbit, the time-delayed feedback control technique for the fractional-order systems is applied, thus controlling the chaotic behaviour.

Conclusions

For fractional-order case, the results showed the influence of derivative order on the dynamics of the AFM system. Due to the fractional order, some phenomena comes up, which were confirmed through detailed numerical investigations by 0–1 test. The time-delayed feedback control technique was efficient to control the chaotic motion of the AFM in fractional order. In addition, the robustness of the proposed time-delayed feedback control was tested by a sensitivity analysis to parametric uncertainties.

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References

  1. Jalili N, Laxminarayana K (2004) A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences. Mechatronics 14:907–945

    Article  Google Scholar 

  2. Korayem MH, Zafari H, Amanati S, Damircheli A, Ebrahimi N (2010) Analysis and control of micro-cantilever in dynamic mode AFM. Int J Adv Manuf Technol 50:979–990

    Article  Google Scholar 

  3. Kuznetsov YG, Malkin A, Lucas R, Plomp M, McPherson A (2001) Imaging of viruses by atomic force microscopy. J Gen Virol 82:2025–2034

    Article  Google Scholar 

  4. Lee SI, Lee JM, Hong SH (2005) Dynamics and control of tapping tip in atomic force microscope for surface measurement applications. CIRP Ann 54:527–530

    Article  Google Scholar 

  5. Salarieh H, Alasty A (2009) Control of chaos in atomic force microscopes using delayed feedback based on entropy minimization. Commun Nonlinear Sci Numer Simul 14:637–644

    Article  Google Scholar 

  6. Karvinen KS, Moheimani SOR (2014) Control of the higher eigenmodes of a microcantilever: applications in atomic force microscopy. Ultramicroscopy 137:66–71

    Article  Google Scholar 

  7. Sadeghpour M, Salarieh H, Alasty A (2013) Controlling chaos in tapping mode atomic force microscopes using improved minimum entropy control. Appl Math Model 37:1599–1606

    Article  MathSciNet  MATH  Google Scholar 

  8. Keyvani A, Alijani F, Sadeghian H, Maturova K, Goosen H, van Keulen F (2017) Chaos: the speed limiting phenomenon in dynamic atomic force microscopy. J Appl Phys 122:224306

    Article  Google Scholar 

  9. Frétigny C (2007) Atomic force microscopy. Nanosci Part I, vol 1, pp 91–119

  10. Garcia R, Pérez R (2002) Dynamic atomic force microscopy method. Surf Sci Rep 47:197–301

    Article  Google Scholar 

  11. Rodrigues KS, Balthazar JM, Tusset AM, Pontes BR, Bueno AM (2014) Preventing chaotic motion in tapping-mode atomic force microscope. J Control Autom Electr Syst 25:732–740

    Article  Google Scholar 

  12. Nozaki R, Balthazar JM, Tusset AM, Pontes BR, Bueno AM (2013) Nonlinear control system applied to atomic force microscope including parametric errors. J Control Autom Electr Syst 24:223–231

    Article  Google Scholar 

  13. Balthazar JM, Bueno AM, Tusset AM, Pontes BR (2012) On an overview of nonlinear and chaotic behavior and their controls of an atomic force microscopy (AFM) vibrating problem. Nonlinearity, Bifurc Chaos - Theory Appl 1:45–68

    Google Scholar 

  14. Burgess W, Daniel S (2006) Chaos affects atomic force microscopes. Photon Spectra 40:101–102

    Google Scholar 

  15. Basso M, Bagni G (2004) Controller synthesis for stabilizing oscillations in tapping-mode atomic force microscopes. In: 2004 IEEE international conference on robotics and automation, New Orleans, LA, USA, 2–4 September 2004. IEEE, pp 372–377

  16. Balthazar JM, Tusset AM, Bueno AM (2014) Nonlinear TM-AFM control considering parametric errors in the control signal evaluation. J Theor Appl Mech (Warsaw) 52:93–106

    Google Scholar 

  17. Zhang WM, Meng G, Zhou JB, Chen JY (2009) Nonlinear dynamics and chaos of microcantilever-based tm-afms with squeeze film damping effects. Sensors 9:3854–3874

    Article  Google Scholar 

  18. Zhang WM, Meng G (2005) Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS. Sens. Actuat. A: Phys. 119:291–299

    Article  Google Scholar 

  19. Zhang WM, Meng G (2007) Nonlinear dynamic analysis of electrostatically actuated resonant MEMS sensors under parametric excitation. IEEE Sens J 7:370–380

    Article  Google Scholar 

  20. Sin CS, Zheng L, Sin JS, Liu F, Liu L (2017) Unsteady flow of viscoelastic fluid with the fractional K-BKZ model between two parallel plates. Appl Math Model 47:114–127

    Article  MathSciNet  MATH  Google Scholar 

  21. Balthazar JM, Tusset AM, De Souza SLT, Bueno AM (2013) Microcantilever chaotic motion suppression in tapping mode atomic force microscope. Proc Inst Mech Eng C 227(8):1730–1741

    Article  Google Scholar 

  22. Bowen WR, Lovitt RW, Wright CJ (2001) Atomic force microscopy study of the adhesion of Saccharomyces cerevisiae. J Colloid Interface Sci 237(1):54–61

    Article  Google Scholar 

  23. Müller DJ, Dufrêne YF (2011) Atomic force microscopy: a nanoscopic window on the cell surface. Tren Cell Biol 21(8):461–469

    Article  Google Scholar 

  24. Möller C, Allen M, Elings V, Engel A, Müller DJ (1999) Tapping-mode atomic force microscopy produces faithful high-resolution images of protein surfaces. Bioph J 77(2):1150–1158

    Article  Google Scholar 

  25. Lamoreaux SK (1997) Demonstration of the Casimir force in the 0.6 to 6 μ m range. Phys Rev Lett 78(1):5–8

    Article  Google Scholar 

  26. Bordag M, Mohideen U, Mostepanenko VM (2001) New developments in the Casimir effect. Phys Rep 353(1–3):1–205

    Article  MathSciNet  MATH  Google Scholar 

  27. Decca RS, López D, Fischbach E, Krause DE (2003) Measurement of the Casimir force between dissimilar metals. Phys Rev Lett 91(5):050402

    Article  Google Scholar 

  28. Bressi G, Carugno G, Onofrio R, Ruoso G (2002) Measurement of the Casimir force between parallel metallic surfaces. Phys Rev Lett 88(4):041804

    Article  Google Scholar 

  29. Hanke A, Schlesener F, Eisenriegler E, Dietrich S (1998) Critical casimir forces between spherical particles in fluids. Phys Rev Lett 81(9):1885

    Article  Google Scholar 

  30. Rutzel S, Lee SI, Raman A (2003) Nonlinear dynamics of atomic-force-microscope probes driven is Lennard-Jones potentials. Proc R Soc Lond 459:1925–1948

    Article  MATH  Google Scholar 

  31. Bagley R, Calico R (1991) Fractional order state equations for the control of viscoelastically damped structures. J Guid Contr Dyn 14:304–311

    Article  Google Scholar 

  32. Yu Y, Li HX, Wang S, Yu J (2009) Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos, Solitons Fractals 42:1181–1189

    Article  MathSciNet  MATH  Google Scholar 

  33. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Netherlands

    MATH  Google Scholar 

  34. Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific, Singapore

    Book  MATH  Google Scholar 

  35. Tusset AM, Balthazar JM, De Lima JJ, Rocha RT, Janzen FC, Yamaguchi PS (2017) On an optimal control applied in atomic force microscopy (AFM) including fractional-order. In: ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2017, Cleveland. Vol 4: 22nd Design for Manufacturing and the Life Cycle Conference; 11th International Conference on Micro- and Nanosystems, p. V004T09A003-10

  36. Dorcak L (2002) Numerical models for the simulation of the fractional-order control systems. arXiv:math/0204108(preprint)

  37. Petráš I (2011) Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science and Business Media, Berlin

    Book  MATH  Google Scholar 

  38. Meral FC, Royston TJ, Magin R (2010) Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15:939–945

    Article  MathSciNet  MATH  Google Scholar 

  39. Gottwald G, Melbourne I (2004) A new test for chaos in deterministic systems. Proc R Soc Lond A 460:603–611

    Article  MathSciNet  MATH  Google Scholar 

  40. Gottwald G, Melbourne I (2005) Testing for chaos in deterministic systems with noise. Physica D 212:100–110

    Article  MathSciNet  MATH  Google Scholar 

  41. Litak G, Bernardini D, Syta A, Rega G, Rysak A (2013) Analysis of chaotic non-isothermal solutions of thermomechanical shape memory oscillators. Eur Phys J Special Topics 222:1637–1647

    Article  Google Scholar 

  42. Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421–428

    Article  Google Scholar 

  43. Yamasue K, Hikihara T (2006) Control of microcantilevers in dynamic force microscopy using time delayed feedback. Rev Sci Instrum 77(5):053703

    Article  Google Scholar 

  44. Bassinello DG, Tusset AM, Rocha RT, Balthazar JM (2018) Dynamical analysis and control of a chaotic microelectromechanical resonator model. Shock Vib 2018:1–10

    Article  Google Scholar 

  45. Tusset AM, Janzen FC, Rocha RT, Balthazar JM (2018) On an optimal control applied in MEMS Oscillator with chaotic behavior including fractional order. Complexity 2018:1–12

    Article  MATH  Google Scholar 

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Acknowledgements

The authors acknowledge support by CNPq, FAPES, FA and CAPES, all Brazilian research funding agencies. In addition, the authors thank the organizing committee of the 14th International Conference on Vibration Engineering and Technology of Machinery (VETOMAC XIV), where part of this work was presented.

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Authors and Affiliations

  1. Federal University of Technology - Paraná, Av Monteiro Lobato, s/n - Km 0, Ponta Grossa, PR, 84016-210, Brazil

    Angelo M. Tusset, Mauricio A. Ribeiro, Wagner B. Lenz, Rodrigo T. Rocha & Jose M. Balthazar

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  1. Angelo M. Tusset
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  2. Mauricio A. Ribeiro
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  3. Wagner B. Lenz
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  4. Rodrigo T. Rocha
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  5. Jose M. Balthazar
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Correspondence to Angelo M. Tusset.

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Tusset, A.M., Ribeiro, M.A., Lenz, W.B. et al. Time Delayed Feedback Control Applied in an Atomic Force Microscopy (AFM) Model in Fractional-Order. J. Vib. Eng. Technol. 8, 327–335 (2020). https://doi.org/10.1007/s42417-019-00166-5

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  • DOI: https://doi.org/10.1007/s42417-019-00166-5

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